Dynamic Foraging Patterns in Antarctic Humpbacks
Ben Weinstein
2017-01-06 13:52:29
## Source: local data frame [12 x 2]
##
## Animal max(timestamp, na.rm = T)
## (int) (time)
## 1 112699 2012-06-17 03:57:31
## 2 121207 2013-05-09 18:49:37
## 3 121208 2013-02-18 07:52:02
## 4 121210 2013-05-05 07:44:41
## 5 123224 2013-05-24 12:13:18
## 6 123232 2013-09-28 07:28:23
## 7 123236 2013-03-18 11:26:09
## 8 131127 2016-07-15 00:58:36
## 9 131130 2016-04-29 17:30:06
## 10 131132 2016-05-10 12:44:39
## 11 131133 2016-07-05 10:55:56
## 12 131136 2016-06-30 10:57:06
0.1 By Month
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
1.3.1 After filitering
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
#for each if 6 argos class observation error
for(x in 1:6){
##argos observation error##
argos_prec[x,1:2,1:2] <- argos_cov[x,,]
#Constructing the covariance matrix
argos_cov[x,1,1] <- argos_sigma[x]
argos_cov[x,1,2] <- 0
argos_cov[x,2,1] <- 0
argos_cov[x,2,2] <- argos_alpha[x]
}
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec[1,1:2,1:2])
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha_mu[state[i,g,t-1],Month[i,g,t]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t],Month[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha_mu[state[i,g,steps[i,g]-1],Month[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec[argos_class[i,g,t,u],1:2,1:2])
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
#for each month
for (m in 1:Months){
#Intercepts
alpha_mu[1,m] ~ dbeta(1,1)
alpha_mu[2,m] ~ dbeta(1,1)
gamma[1,m] ~ dbeta(3,2) ## gamma for state 1
dev[m] ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2,m] <- gamma[1,m] * dev[m]
}
##Behavioral States
#Hierarchical structure across motnhs
#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos precision, from Jonsen 2005, 2016, represented as precision not sd
#by argos class
argos_sigma[1] <- 11.9016
argos_sigma[2] <- 10.2775
argos_sigma[3] <- 1.228984
argos_sigma[4] <- 2.162593
argos_sigma[5] <- 3.885832
argos_sigma[6] <- 0.0565539
#latitidunal argos precision, from Jonsen 2005, 2016
argos_alpha[1] <- 67.12537
argos_alpha[2] <- 14.73474
argos_alpha[3] <- 4.718973
argos_alpha[4] <- 0.3872023
argos_alpha[5] <- 3.836444
argos_alpha[6] <- 0.1081156
}"
,fill=TRUE)sink()
## user system elapsed
## 269.989 2.380 37110.5731.4 Chains
## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1486124 79.4 3886542 207.6 3886542 207.6
## Vcells 265483546 2025.5 537136906 4098.1 458856903 3500.9## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1340845 71.7 3886542 207.6 3886542 207.6
## Vcells 43972458 335.5 429709524 3278.5 458856903 3500.9
1.5 Change in autocorrelation over time
2 Change in transition probabilities over time
2.1 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1,1] 0.80939346 0.724762645 0.88613117
## 2 alpha_mu alpha_mu[2,1] 0.17571475 0.102955854 0.27153681
## 3 alpha_mu alpha_mu[1,2] 0.87632760 0.779557575 0.94156395
## 4 alpha_mu alpha_mu[2,2] 0.12880976 0.069129913 0.22523318
## 5 alpha_mu alpha_mu[1,3] 0.87926173 0.797097442 0.95165829
## 6 alpha_mu alpha_mu[2,3] 0.07275752 0.035190811 0.12449611
## 7 alpha_mu alpha_mu[1,4] 0.74446667 0.599963292 0.89971100
## 8 alpha_mu alpha_mu[2,4] 0.17960882 0.080108656 0.31319005
## 9 alpha_mu alpha_mu[1,5] 0.75987608 0.578890583 0.90243776
## 10 alpha_mu alpha_mu[2,5] 0.08120333 0.033314499 0.14775759
## 11 gamma gamma[1,1] 0.92798009 0.877399400 0.97385910
## 12 gamma gamma[2,1] 0.15202233 0.030398251 0.30095090
## 13 gamma gamma[1,2] 0.82027378 0.758618330 0.89599102
## 14 gamma gamma[2,2] 0.10001532 0.007260477 0.25272041
## 15 gamma gamma[1,3] 0.77875245 0.695370754 0.86984580
## 16 gamma gamma[2,3] 0.07527076 0.003282894 0.19242231
## 17 gamma gamma[1,4] 0.88472363 0.760832949 0.97518770
## 18 gamma gamma[2,4] 0.09182517 0.007940480 0.25261667
## 19 gamma gamma[1,5] 0.72704760 0.571240546 0.88764601
## 20 gamma gamma[2,5] 0.39217730 0.202489244 0.56002231
## 21 theta theta[1] 0.02488821 0.007324217 0.04235485
## 22 theta theta[2] 3.08718613 2.785248431 3.37649120
3 Behavioral Prediction
3.1 Spatial Prediction
3.2 By individual
3.3 Autocorrelation in behavior
3.3.1 As single timeline
4 Simulated tracks
4.1 Behavioral description
4.1.1 Proportion of states by month
4.1.2 Distance between bouts
5 Behavior duration
## Source: local data frame [140 x 7]
## Groups: Animal, Track, Bout, phistate [140]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 2 1 Traveling 1 0.4781829 January
## 2 1 4 1 Traveling 1 1.5586921 January
## 3 1 5 1 Traveling 1 0.6500116 January
## 4 1 7 1 Traveling 2 0.9866435 February
## 5 1 10 1 Traveling 2 0.9215972 February
## 6 1 11 1 Traveling 2 2.6218171 February
## 7 2 1 1 Traveling 1 16.1553009 January
## 8 2 1 3 Traveling 2 14.1499421 February
## 9 2 2 1 Traveling 3 8.6612037 March
## 10 2 2 3 Traveling 3 12.9640741 March
## .. ... ... ... ... ... ... ...## Source: local data frame [172 x 7]
## Groups: Animal, Track, Bout, phistate [172]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Area-restricted Search 1 0.6059028 January
## 2 1 2 2 Area-restricted Search 1 0.0000000 January
## 3 1 3 1 Area-restricted Search 1 0.6483912 January
## 4 1 6 1 Area-restricted Search 1 2.6031250 January
## 5 1 8 1 Area-restricted Search 2 0.5460301 February
## 6 1 9 1 Area-restricted Search 2 3.3496065 February
## 7 1 12 1 Area-restricted Search 2 0.6120602 February
## 8 2 1 2 Area-restricted Search 2 0.1554977 February
## 9 2 2 2 Area-restricted Search 3 10.6119097 March
## 10 2 2 4 Area-restricted Search 4 2.4937847 April
## .. ... ... ... ... ... ... ...## Source: local data frame [312 x 7]
## Groups: Animal, Track, Bout, phistate [312]
##
## Animal Track Bout phistate MonthF Days Month
## (dbl) (dbl) (int) (fctr) (dbl) (dbl) (fctr)
## 1 1 1 1 Area-restricted Search 1 0.6059028 January
## 2 1 2 1 Traveling 1 0.4781829 January
## 3 1 2 2 Area-restricted Search 1 0.0000000 January
## 4 1 3 1 Area-restricted Search 1 0.6483912 January
## 5 1 4 1 Traveling 1 1.5586921 January
## 6 1 5 1 Traveling 1 0.6500116 January
## 7 1 6 1 Area-restricted Search 1 2.6031250 January
## 8 1 7 1 Traveling 2 0.9866435 February
## 9 1 8 1 Area-restricted Search 2 0.5460301 February
## 10 1 9 1 Area-restricted Search 2 3.3496065 February
## .. ... ... ... ... ... ... ...
5.1 Proportion of time allocation
## Month Traveling Area-restricted Search PropF TotalTime
## 1 January 124.81784 152.41347 0.5497701 277.23131
## 2 February 230.64413 338.60061 0.5948243 569.24475
## 3 March 98.93443 207.06300 0.6766821 305.99743
## 4 April 50.42377 163.84080 0.7646658 214.26457
## 5 May 22.92902 60.52991 0.7252659 83.458925.2 Number of bouts
6 Time spent in grid cell
6.1 All years
6.2 Add in sea ice
